The product integral
This is a survey paper on the product integral, which is defined as \(\int_ a^{-b}(E+f(t)dt)=\lim\prod(f,T)\), where \(\prod(f,T)=(E+f(t_ 0)\Delta t_ 0)(E+f(t_ 1)\Delta t_ 1)\ldots(E+f(t_ n)\Delta t_ n)\), \(t_ 0,t_ 1,\ldots,t_ n\in[a,b]\), and the limit is taken when \(\max(t_ i-t_{i-1})\to 0\), \(T\) being a partition of the interval \([a,b]\). The most important property of \(F(x)=\int^{-x}_ a(E+f(t)dt)\) is \(F'(x)=F(x)f(x)\). Then the product integral of a system of linear differential equations that of a system of differential operators, the curvilinear product integral, the curvature of such a curvilinear product integral the potential of a curvilinear product integral of zero curvature are considered. The case when \(f\) is a polynomial is studied, under the condition that \(F\) is also a polynomial. The study of orbits in the space of polynomial matrices is a final paragraph. Applications to geometry, especially for functionally abelian connections in Riemannian spaces.
orbits in the space of polynomial matrices
functionally abelian connections in Riemannian spaces
scientific article; zbMATH DE number 25528